The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions

Abstract

<p style='text-indent:20px;'>The small and large size behavior of stationary solutions to the fragmentation equation with size diffusion is investigated. It is shown that these solutions behave like stretched exponentials for large sizes, the exponent in the exponential being solely given by the behavior of the overall fragmentation rate at infinity. In contrast, the small size behavior is partially governed by the daughter fragmentation distribution and is at most linear, with possibly non-algebraic behavior. Explicit solutions are also provided for particular fragmentation coefficients.</p>

Analytical cell size distribution: lineage-population bias and parameter inference

“…In the corresponding Langevin representation of the cell cycle, size grows as dx=νfalse(xfalse) dt+2D(x) dW, where W is the Wiener process. Equation (2.1) reduces to a growth-fragmentation equation when single-cell growth is deterministic ( D ( x ) = 0) [16], to a diffusion-fragmentation equation when there is no single-cell growth ( ν ( x ) = 0) [25,26] and to a fragmentation equation when D ( x ) = 0 and ν ( x ) = 0 [27].…”

“…Equation (2.1) reduces to a growth-fragmentation equation when single-cell growth is deterministic (D(x) = 0) [16], to a royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20220405 diffusion-fragmentation equation when there is no single-cell growth (ν(x) = 0) [25,26] and to a fragmentation equation when D(x) = 0 and ν(x) = 0 [27].…”